Optimal. Leaf size=78 \[ -\frac{p (a d-b e)^2 \log (a x+b)}{2 a^2 e}+\frac{(d+e x)^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{2 e}+\frac{b e p x}{2 a}+\frac{d^2 p \log (x)}{2 e} \]
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Rubi [A] time = 0.0561594, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2463, 514, 72} \[ -\frac{p (a d-b e)^2 \log (a x+b)}{2 a^2 e}+\frac{(d+e x)^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{2 e}+\frac{b e p x}{2 a}+\frac{d^2 p \log (x)}{2 e} \]
Antiderivative was successfully verified.
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Rule 2463
Rule 514
Rule 72
Rubi steps
\begin{align*} \int (d+e x) \log \left (c \left (a+\frac{b}{x}\right )^p\right ) \, dx &=\frac{(d+e x)^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{2 e}+\frac{(b p) \int \frac{(d+e x)^2}{\left (a+\frac{b}{x}\right ) x^2} \, dx}{2 e}\\ &=\frac{(d+e x)^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{2 e}+\frac{(b p) \int \frac{(d+e x)^2}{x (b+a x)} \, dx}{2 e}\\ &=\frac{(d+e x)^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{2 e}+\frac{(b p) \int \left (\frac{e^2}{a}+\frac{d^2}{b x}-\frac{(a d-b e)^2}{a b (b+a x)}\right ) \, dx}{2 e}\\ &=\frac{b e p x}{2 a}+\frac{(d+e x)^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{2 e}+\frac{d^2 p \log (x)}{2 e}-\frac{(a d-b e)^2 p \log (b+a x)}{2 a^2 e}\\ \end{align*}
Mathematica [A] time = 0.0269028, size = 85, normalized size = 1.09 \[ \frac{1}{2} b e p \left (\frac{x}{a}-\frac{b \log (a x+b)}{a^2}\right )+d x \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{1}{2} e x^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{b d p \log \left (a+\frac{b}{x}\right )}{a}+\frac{b d p \log (x)}{a} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.109, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) \ln \left ( c \left ( a+{\frac{b}{x}} \right ) ^{p} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02641, size = 74, normalized size = 0.95 \begin{align*} \frac{1}{2} \, b p{\left (\frac{e x}{a} + \frac{{\left (2 \, a d - b e\right )} \log \left (a x + b\right )}{a^{2}}\right )} + \frac{1}{2} \,{\left (e x^{2} + 2 \, d x\right )} \log \left ({\left (a + \frac{b}{x}\right )}^{p} c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68494, size = 184, normalized size = 2.36 \begin{align*} \frac{a b e p x +{\left (2 \, a b d - b^{2} e\right )} p \log \left (a x + b\right ) +{\left (a^{2} e x^{2} + 2 \, a^{2} d x\right )} \log \left (c\right ) +{\left (a^{2} e p x^{2} + 2 \, a^{2} d p x\right )} \log \left (\frac{a x + b}{x}\right )}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.9264, size = 156, normalized size = 2. \begin{align*} \begin{cases} d p x \log{\left (a + \frac{b}{x} \right )} + d x \log{\left (c \right )} + \frac{e p x^{2} \log{\left (a + \frac{b}{x} \right )}}{2} + \frac{e x^{2} \log{\left (c \right )}}{2} + \frac{b d p \log{\left (x + \frac{b}{a} \right )}}{a} + \frac{b e p x}{2 a} - \frac{b^{2} e p \log{\left (x + \frac{b}{a} \right )}}{2 a^{2}} & \text{for}\: a \neq 0 \\d p x \log{\left (b \right )} - d p x \log{\left (x \right )} + d p x + d x \log{\left (c \right )} + \frac{e p x^{2} \log{\left (b \right )}}{2} - \frac{e p x^{2} \log{\left (x \right )}}{2} + \frac{e p x^{2}}{4} + \frac{e x^{2} \log{\left (c \right )}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26641, size = 151, normalized size = 1.94 \begin{align*} \frac{a^{2} p x^{2} e \log \left (a x + b\right ) - a^{2} p x^{2} e \log \left (x\right ) + 2 \, a^{2} d p x \log \left (a x + b\right ) + a^{2} x^{2} e \log \left (c\right ) - 2 \, a^{2} d p x \log \left (x\right ) + a b p x e + 2 \, a b d p \log \left (a x + b\right ) - b^{2} p e \log \left (a x + b\right ) + 2 \, a^{2} d x \log \left (c\right )}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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